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This paper presents an investigation on nonlinear forced vibration characteristics of a spinning shaft‐disk assembly resting on sliding bearing supports. Based on the Timoshenko beam theory and Kirchhoff plate theory, the coupled model of the rotor is established. The present model is validated through comparison with the experiment results for different structure parameters. Considering the gyroscopic effect and nonlinear oil film force, the equations of motion are derived by employing the Lagrange equation. Then, the modal synthesis method and the Runge‐Kutta method are adopted to obtain forced vibration responses of the spinning assembly. Special attention is paid to examine the effect of spinning speed on the nonlinear vibration behaviors of the shaft‐disk rotor system, which can play an especially important role in an actual project. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of infinitely many solutions of the following degenerate magnetic fractional problem involving critical Sobolev‐Hardy nonlinearities: M([u]s,A2)(−△)Asu=λf(x,|u|)u+|u|2s∗(α)−2u|x|α,inRN,where s ∈ (0,1), N > 2s, 2s∗(α)=2(N−α)N−2s is the fractional Hardy‐Sobolev critical exponent with α ∈ [0,2s), λ is a positive real parameter, M:R0+→R0+ is a Kirchhoff function, A:RN→RN is a magnetic potential function, and (−△)As is the fractional magnetic operator. By using the new version of symmetric mountain pass theorem of Kajikiya, we prove that the problem admits infinitely many solutions for the suitable value of λ. [ABSTRACT FROM AUTHOR]

We consider a class of variational systems involving fractional Kirchhoff‐type equations of the form: M1(‖u‖X2)(−Δ)su=Fu(x,u,v)inΩ,M2(‖v‖X2)(−Δ)sv=Fv(x,u,v)inΩ,u=v=0inRN\Ω,where s ∈ (0,1), N > 2s, Ω⊂RN a smooth and bounded domain, the functions Fu, Fv, M1 and M2 are continuous and ( − Δ)s is the fractional Laplacian operator. In this paper, we show that, under appropriate growth conditions on the nonlinearities Fu and Fv and on the nonnegative functions M1 and M2, the (weak) solutions are precisely the critical points of a related functional defined on a fractional Hilbert space Y(Ω) = X(Ω) × X(Ω) and the existence infinitely many solutions can be obtained by the use of the Krasnoselskii's genus. Besides, a regularity result can also be obtained by using specific results for systems in conjunction with the growth assumptions of these functions. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and uniqueness of global solutions and uniform stabilization of the energy of initial‐boundary value problems for quasilinear equations of the Kirchhoff type. The asymptotic stabilization of the solutions of these problems are obtained without any dissipative mechanism acting on the displacement variable. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and concentration behavior of positive solutions for the following Kirchhoff type equation: − ( ɛ 2 a + ɛ b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = K ( x ) u p , x ∈ R 3 , u ∈ H 1 ( R 3 ) , u ( x ) > 0 , where ɛ is a positive parameter, a and b are positive constants, and 3 0 small. Furthermore, we show that the solution uɛ concentrates at the minimum points of A ( s ) as ɛ→0+. [ABSTRACT FROM AUTHOR]

In this paper, the existence and multiplicity of positive solutions are obtained for a class of Kirchhoff type problems with two singular terms and sign‐changing potential by the Nehari method. [ABSTRACT FROM AUTHOR]

A semilinear parabolic problem is considered in a thin 3-D star-shaped junction that consists of several thin curvilinear cylinders that are joined through a domain (node) of diameter [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of ground state sign-changing solutions for the following class of elliptic equation: [ABSTRACT FROM AUTHOR]

In this paper, we investigate the influence of boundary dissipation on the decay property of solutions for a transmission problem of Kirchhoff-type wave equations with a memory condition on one part of the boundary. Without the condition u0 = 0 on Γ0, we establish a general decay of energy depending on the behavior of relaxation function by introducing suitable energy and Lyapunov functionals. This result allows a wider class of relaxation functions. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Consider the following fractional Kirchhoff equations involving critical exponent: [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of positive solutions to a class of nonlocal boundary value problem of the p-Kirchhoff type [ABSTRACT FROM AUTHOR]

This paper is concerned with the following systems of Kirchhoff-type equations: [ABSTRACT FROM AUTHOR]

In this paper, we study a viscoelastic wave equations of Kirchhoff type with acoustic boundary conditions. For a viscoelastic wave equations of Kirchhoff type without strong damping, we prove an explicit and general decay rate result, using some properties of the convex functions. For a wider class of relaxation functions, we establish a more general decay result, from which the usual exponential and polynomial decay rates are only special cases. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we consider the coupled system of Kirchhoff-type equations: [ABSTRACT FROM AUTHOR]

The aim of this note is to investigate the existence of signed and sign-changing solutions to the Kirchhoff type problem [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of solutions to a class of p( x)-Kirchhoff-type equations with Dirichlet boundary data as follows: By means of variational methods and the theory of the variable exponent Sobolev spaces, we establish some conditions on the existence of solutions. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We derive a linearized prestressed elastic shellmodel froma nonlinear Kirchhoff model of elastic plates. The model is given in terms of displacement and micro-rotation of the cross-sections. In addition to the standardmembrane, transverse shear, and flexural terms, the model also contains a nonstandard prestress term. The prestress is of the same order as flexural effects; hence, themodel is appropriate when flexural effects dominate overmembrane ones. We prove the existence and uniqueness of the solutions by Lax-Milgramtheorem and compare solution with the solution of the standard shellmodel via numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence and non-existence of positive solutions for p-Kirchhoff type problems with a parameter on [ABSTRACT FROM AUTHOR]

In this paper, we study the following modified Kirchhoff-type equations of the form: [ABSTRACT FROM AUTHOR]

In this paper, we study the following modified Kirchhoff-type equations of the form: [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the following nonlinear problem of Kirchhoff type with critical Sobolev exponent: − a + b ∫ R 3 | Du | 2 Δu + u = f ( x , u ) + u 5 , x ∈ R 3 , u ∈ H 1 ( R 3 ) , u > 0 , x ∈ R 3 , where a, b > 0 are constants. Under certain assumptions on the sign‐changing function f(x,u), we prove the existence of positive solutions by variational methods. Our main results can be viewed as a partial extension of a recent result of He and Zou in [Journal of Differential Equations, 2012] concerning the existence of positive solutions to the nonlinear Kirchhoff problem − ϵ 2 a + ϵb ∫ R 3 | Du | 2 Δu + V ( x ) u = f ( u ) , x ∈ R 3 , u ∈ H 1 ( R 3 ) , u > 0 , x ∈ R 3 , where ϵ > 0 is a parameter, V (x) is a positive continuous potential, and f ( u ) ∽ | u | p − 2 u with 4 < p < 6 and satisfies the Ambrosetti–Rabinowitz type condition. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]